Nonvanishing boundary conditions and dark solitons in the NLS model

The convenient form of the NLS for dealing with non-vanishing boundary conditions, which support dark-soliton like solutions [6], can be written as

This form of NLS model is supplied with the non-vanishing boundary conditions given by [1,2,3]

In order to give a group theoretical construction of the system above, let us consider the Lax pair A and

where Ψ ± , ϕ 1 and ϕ 2 are the fields of the sl( 2) NLS model and the potential A and B lie in the sl(2) affine Kac-Moody Lie algebra. The Lax pair in (0.3)-(0.4) provided with the zero-curvature condition

furnishes the model (0.1) provided that the following transformation t → it, x → ix, Ψ ± → Ψ ± ǫ ∓2 , and the identification ψ ≡ Ψ + = (Ψ -) * , are made. The factor ǫ ∓2 is introduced for later convenience and * means complex conjugation. It was considered Ψ + 0 Ψ - 0 = -ρ 2 . The vacuum solutions to be considered are the ones of constant configuration, Ψ ± 0 = ρǫ ∓2 , ϕ 1 = ϕ 2 = 0; so, the vacuum fields A V and B V from (0.3)-(0.4) are

The vacuum connections can be written in the form

where Ψ is the group element

with k ± being constants, I the identity matrix and E ± , σ 3 Pauli matrices. The connections in (0.7) are called pure gauge solutions and are solutions of the zero-curvature condition (0.5).

Considering a 2 × 2 matrix representation for (0.8) in sl(2) algebra, it is possible to write certain relationships between the parameters k + , k -, ζ, κ and λ

The relationships in (0.9) show that ζ and κ assume two possible values in terms of λ, this requires the construction of Riemann sheets.

Here it is introduced an affine parameter ξ such that the functions

(0.10) become single valued in terms of ξ. The function ζ(ξ) above is known as the Zukowsky function.

The appearance of an affine parameter motivates us to introduce a new spectral parameter associated with the potentials  and

The potentials (0.11) and (0.12) written in terms of the new spectral parameter ξ describe the NLS model (0.1) when the zero-curvature condition (0.5) is used. So, the vacuum connections corresponding to (0.11) and (0.12) are given by

Notice that these potentials are deformations of the ones in (0.6).

In terms of the new spectral parameter ξ, (0.8) takes the form

where

When ρ → 0 one has ξ → λ, which implies that P → I.

1 The dressing transformation

The dressing transformations are non-local gauge transformations that act on the fields of the model preserving their gradation structure; they are made with the aid of two group elements Θ + and Θ -, such that

It is assumed the generalized Gauss decomposition

The vector tau function τ (x, t) is defined by [7] τ

Once the highest weight state | λ0 is an eigenstate of g 0 subalgebra, it is possible to define

where τ0 (x, t) is a function described by

Using (1.3) and (1.5) one finds

Replacing the fields A V and B V , in the form given in (0.6), into the dressing transformation (1.1), one gets

where Θ -= exp n>0 σ-n , M = exp(σ 0 ) and N = exp n>0 σn . It is possible to find some of the components, say σ-1 , σ-2 , in terms of the fields Ψ ± , ϕ 1 and

So, with the aid of (1.6) the solutions in the orbit of the vacuum are given by

where the τ ± functions are defined by

According to the dressing method, the soliton solutions are determined by choosing convenient constant group elements h.

Let us choose the group element h = e F , F = ∞ n=-∞ ν n 1 E -n -; the relevant τ functions are

τ -= e -ϕ1 λ0 |E 1 + (P F P -1 )| λ0 ;

(1.16) so the equations (1.11) provided the matrix elements in (1.14), (1.15) and (1.16), furnishes the solution

(1.17)

where a 1 is a free parameter and

. This is just the one dark-soliton solution. The relevant matrix elements can be obtained with the aid of the one level vertex operator or the integrable highest weight representations of the sl(2) Kac-Moody algebra. Similarly, one can set h = e G , where

In this way one can get another one dark-soliton solution. In order to get insight into the ‘dark’-soliton evolution let us plot the function (Ψ + Ψ -) for two successive times. The figure 1 is

In order to obtain 2-soliton solution it is chosen h = e F e G . The relevant τ functions become

As usual the matrix elements above can be computed through the relevant highest weight representation of the affine Lie algebra, however one can avoid those calculations by writing these matrix elements as certain constant parameters which must be de

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